As we begin to explore geometric invariant theory (GIT), it will be useful to establish a sturdy foundation off of which we can build. In this talk, we will cover some of the fundamental concepts of algebraic geometry that will be important to understand the deeper theory to come. We will start by defining affine and projective varieties, some of the main objects of study in algebraic geometry, and we will work our way to establishing morphisms of varieties. This talk is meant to be as accessible as possible and assumes no prerequisite knowledge other than a very basic understanding of rings and ideals.

Category Theory

15 Apr:Jules Hedges- Open Games: The long road to practical applications

Abstract: I will talk about open games, and the closely related concepts of lenses/optics and open learners. My goal is to report on the successes and failures of an ongoing effort to try to realise the often-claimed benefits of categories and compositionality in actual practice. I will introduce what little theory is needed along the way. Here are some things I plan to talk about:

Abstract: Symmetric monoidal categories with duals, a.k.a. compact monoidal categories, have a pleasing string diagram calculus. In particular, any compact monoidal category is closed with [A,B] = (A* ⊗ B), and the transpose of A ⊗ B → C to A → [B,C] is represented by simply bending a string. Unfortunately, a closed symmetric monoidal category cannot even be embedded fully-faithfully into a compact one unless it is traced; and while string diagram calculi for closed monoidal categories have been proposed, they are more complicated, e.g. with "clasps" and "bubbles". In this talk we obtain a string diagram calculus for closed symmetric monoidal categories that looks almost like the compact case, by fully embedding any such category in a star-autonomous one (via a functor that preserves the closed structure) and using the known string diagram calculus for star-autonomous categories. No knowledge of star-autonomous categories will be assumed.

29 Apr:Gershom Bazerman- A localic approach to the semantics of dependency, conflict, and concurrency

Abstract: Petri nets have been of interest to applied category theory for some time. Back in the 1980s, one approach to their semantics was given by algebraic gadgets called "event structures." We use classical techniques from order theory to study event structures without conflict restrictions (which we term "dependency structures with choice") by their associated "traces", which let us establish a one-to-one correspondence between DSCs and a certain class of locales. These locales have an internal logic of reachability, which can be equipped with "versioning" modalities that let us abstract away certain unnecessary detail from an underlying DSC. With this in hand we can give a general notion of what it means to "solve a dependency problem" and combinatorial results bounding the complexity of this. Time permitting, I will sketch work-in-progress which hopes to equip these locales with a notion of conflict, letting us capture the full semantics of general event structures in the form of homological data, thus providing one avenue to the topological semantics of concurrent systems. This is joint work with Raymond Puzio.

Organizers

Upcoming Talk

24 Jan 2020

Correspondence between Dominated Splitting and Spectrum of the Jacobi Operator

Kateryna Alkorn

We will talk about one of the basic results relating spectrum of Schrodinger operator and Uniform Hyperbolicity of the associated Schrodinger cocycle. After that we will proceed by extending the existing theory into a more general setting relating Dominated Splitting and Spectrum of Jacobi operator. We will talk about the problems we have encountered along the way and possible ways to resolve them. In the end we will see a proof of partial result from our hypothesis.

Scheduled Talks

31 Jan 2020

Title TBD

William Hoffer

24 Jan 2020

Correspondence between Dominated Splitting and Spectrum of the Jacobi Operator

Kateryna Alkorn

We will talk about one of the basic results relating spectrum of Schrodinger operator and Uniform Hyperbolicity of the associated Schrodinger cocycle. After that we will proceed by extending the existing theory into a more general setting relating Dominated Splitting and Spectrum of Jacobi operator. We will talk about the problems we have encountered along the way and possible ways to resolve them. In the end we will see a proof of partial result from our hypothesis.

17 Jan 2020

Upper bounds of the cop number

Raymond Matson

The game of cops and robbers is a type of graph searching problem where a team of cops try to capture a robber by moving onto the same vertex as the robber. The canonical question that arises is: what is the smallest number of cops needed to ensure that the cops will win for any graph of order 𝑛? Henri Meyniel conjectured that for any connected graph of order 𝑛, the number of cops needed is 𝑂(𝑛−−√). We will explore the upper bound of some specific graphs as well as attempts to prove Meyniel's conjecture.

10 Jan 2020

An Introduction to Interval Bundles

Jonathan Alcaraz

Interval bundles are a way of taking a space we know and making it more topologically complicated while still maintaining homotopy type and hence maintaining the fundamental group. In low-dimensional topology, we see these when studying 3-manifolds. It turns out that if a 3-manifold smells like a surface (ie, its fundamental group is a surface group), then it is an interval bundle over a surface.

Fall 2019

6 Dec 2019

Department Potluck!

29 Nov 2019

No Meeting

22 Nov 2019

On the Stability of Self-Similar Blowup in Nonlinear Wave Equations

Michael McNulty

When studying nonlinear wave equations, one concerns themselves with the well-posedness of the Cauchy problem. Does a solution exist for some amount of time? Does it exist for all time? Is it unique? Does the solution depend continuously on the initial data? Within the context of energy supercritical wave equations, a typical way for solutions to fail to exist for all time is through the phenomenon of self-similar blowup. After making this observation, one is left pondering the stability of this blowup. In other words, one wants to know if there is an entire open set of initial data leading to this blowup. Answering this question for particular wave equations is an active area of research with lots of techniques stemming from wave maps. In this talk, we will discuss current work in progress toward establishing the asymptotic nonlinear stability of self-similar blowup in the strong-field Skyrme model.

15 Nov 2019

Fractions, Continued: A Look at Continued Fractions

Nick Newsome

Continued fractions have been the subject of study for many years. A continued fraction is an expression that iteratively describes any real number. Rational numbers have a finite continued fraction representation, while irrationals have an infinite continued fraction representation. Continued fractions have applications ranging from approximating real numbers to solving Diophantine equations to (as I discovered recently) the study of differential equations. Although they are not incredibly difficult to comprehend, continued fractions are not typically part of the standard undergraduate (or graduate) mathematics curriculum. To that end, this talk will endeavor to introduce the basics of continued fractions in the hopes of showing how they can be used to do some pretty cool stuff. For example, have you ever considered just how irrational an irrational number can be? Continued fractions give us a way to develop a sort of hierarchy of irrationality. We will also (hopefully) show how continued fractions can be used to solve a particular Diophantine equation.

8 Nov 2019

No Meeting

1 Nov 2019

The Variable, Free and Bound

Christian Williams

In mathematics, computer science, and logic, one of the most useful ideas is also the most humble: the variable. But what exactly is a variable? When we write f(x)=x+3, how do we formalize the distinction between the "placeholder" x to be substituted, and the "real" value 3? Conventional algebra and logic do not answer this question; we simply take variables for granted. Though not widely known, the answer was given 25 years ago, in a paper called "Abstract Syntax with Variable Binding". This summer I got to visit the mathematician who realized this idea, and began to join in a grand project of laying the algebraic foundations of formal languages.
Essentially, it is a new take on the ideas I presented last year: rather than thinking of algebraic theories as categories, one can think of them as functors T:C->Set, from a category of contexts to the category of sets -- one interprets T(c) as the set of terms which can be derived from a context c (for simple languages, C=N, and a context "n" simply represents having n free variables). From this perspective, one can reformulate all of ordinary algebra; but this "category of presheaves" has a richer structure, in which we can say and do much more: in particular, we can formalize variables in a natural way. This is the key to having a universal language in which to express all of those we use in math and programming.
Join me as we explore the beautiful world of "functors as languages", where we will connect such different concepts as the lambda calculus, species and simplicial sets, algebra and logic -- there will even be integrals. Hope to see you there.

25 Oct 2019

A Look at Microlocal Analysis

Michael McNulty

In the study of differential equations, one is always interested in studying equations of the form 𝑃𝑢=𝑓 where 𝑢 is some unknown function, 𝑓 is some known function, and 𝑃 is some differential operator. Typically, one hopes to solve for the unknown 𝑢 or to extract necessary properties of it. In other words, one wants to make sense of the right-hand side of 𝑢=𝑃−1𝑓 or to know what spaces 𝑢 could live in. So, what does it mean to invert a differential operator? Where could the solution possibly live? Microlocal analysis provides a framework in which one can answer these questions. A good place to start the descent into microlocal analysis is with the study of pseudodifferential operators and how they propagate singularities. We will see how singularities of a distribution propagate along particular directions in the cotangent bundle which are determined uniquely by the pseudodifferential operator acting on them.

18 Oct 2019

Finitely Additive Invariant Set Functions and Paradoxical Decompositions, or: How I Learned to Stop Worrying and Love the Axiom of Choice

Adam D. Richardson

This talk introduces the historic 𝜎-additive measure problem in ℝ𝑛 and describes how the existence of nonmeasurable sets provided an answer to this problem that led mathematicians to explore the consequent finitely additive measure problem in ℝ𝑛. The Axiom of Choice plays an inextricable role in these problems. The existence of a finitely additive measure on 𝑆1 is developed carefully using results from functional analysis before the problem is explored in general. The application of the Axiom of Choice in these problems can yield paradoxical decompositions of subsets of ℝ𝑛 (and by extension ℝ𝑛 itself) such as the seminal Hausdorff half-third paradox as well as the eponymous Banach-Tarski paradox. The development of these paradoxes is group theoretic in nature, and some of the group properties which yield such decompositions are discussed. This talk seeks to tell the mathematical origin story of such paradoxes, including detailing the Hausdorff half-third paradox, while highlighting how the controversial Axiom of Choice led to these wholly counterintuitive yet absolutely fascinating measure-theoretic results.

11 Oct 2019

Introduction to Operads

Joe Moeller

I'll give an intuitive introduction to the notion of "operad", a categorical tool for describing algebraic structures. Then we'll look at a few of the main examples that people use in homotopy theory and combinatorics. Time permitting, I'll also talk about the recognition principle and Kozsul duality.

4 Oct 2019

A Crash Course on 𝑞-Calculus

Michael Pierce

So 𝑞-calculus, also called quantum calculus, by itself is just a "generalization" of arithmetic and calculus. However some arithmetic factoids in 𝑞-calculus come up in representation theory and mathematical physics research. The goal of this talk though is to introduce 𝑞-calculus as a stand-alone topic, and to familiarize the audience with some of those factoids so that it won't be too jarring to them if those factoids pop up in research. If time permits I'll also talk about current research being done into pure 𝑞-calculus, and explain a bit about why that word generalization above is in quotes.

27 Sep 2019

Interpreting Mathematics Rigorously

Alexander Martin

Rigor is the language of mathematics, but currently our languages are inherently not rigorous. It is possible for a student, using a pencil on paper with the notations and terms presented in the class, to make an error in a mathematical statement. I have been working on a project to develop a language which accounts for mathematical rigor. One major design goal is that assumptions, claims, and implication relations come from making a statement as opposed to from reading it, making it impossible to state something incorrect. The language is explicitly compiled by a computer so it makes sense to distinguish between valid statements and nonsense, nonsense being something which throws an error upon compilation. Any statement which can be stated (i.e. successfully compiled) is then tautologically true by design, so we will see how this works and what it means.
Another goal is to be able to write new definitions, make new claims, and prove them in a way which a computer can understand and without modifying the language itself. This language involves interpreting mathematical statements as directed acyclic graphs, so we will see how that works. Statements are saved in XML files (like SVG and XHTML if you have ever seen those) and I have written a compiler in javascript (web browser). This talk is an overview of what I have developed so far, both the language itself and the tools to interact with it, and what I hope the future holds for the project.